Limits, Continuity and Differentiability PYQs - Last 5 Years
MHT CET / Mathematics / Calculus / 129 recent questions
MathematicsCalculus2022-2026
Practice 129 MHT CET Mathematics questions from Limits, Continuity and Differentiability. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.
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Last 5 Years Limits, Continuity and Differentiability Questions
Showing 29 of 129 filtered questions.
1Limits Continuity And Differentiability
If $\mathrm{f}(x)=\left(\frac{1+\tan x}{1+\sin x}\right)^{\operatorname{cosec} x}$ is continuous at $x=0$ then $f(0)$ is equal to
MCQ+2 / -02024
2Limits Continuity And Differentiability
The number of discontinuities of the greatest integer function \(\mathrm{f}(x)=[x], x \in\left(-\frac{7}{2}, 100\right)\)
MCQ+2 / -02023
3Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^3}\) equals
MCQ+2 / -02023
4Limits Continuity And Differentiability
$$f(x)=\left\{\begin{array}{ll}
\frac{1-\cos k x}{x^2}, & \text { if } x \leq 0 \\
\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text { if } x>0
\end{array}\right. \text { is continuous at }$$
\(x=0\), then the value of \(\mathrm{k}\) is
\(x=0\), then the value of \(\mathrm{k}\) is
MCQ+2 / -02023
5Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 0} \frac{(1-\cos 2 x) \cdot \sin 5 x}{x^2 \sin 3 x}\) is
MCQ+2 / -02023
6Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=\)
MCQ+2 / -02023
7Limits Continuity And Differentiability
Let \(\mathrm{f}(x)=5-|x-2|\) and \(\mathrm{g}(x)=|x+1|, x \in \mathrm{R}\) If \(\mathrm{f}(x)\) attains maximum value at \(\alpha\) and \(\mathrm{g}(x)\) attains minimum value at \(\beta\), then $$\lim _\limits{x \rightarrow-\alpha \beta} ...
MCQ+2 / -02023
8Limits Continuity And Differentiability
If \(f(a)=2, f^{\prime}(a)=1, g(a)=-1, g^{\prime}(a)=2\), then as \(x\) approaches a, \(\frac{\mathrm{g}(x) \mathrm{f}(\mathrm{a})-\mathrm{g}(\mathrm{a}) \mathrm{f}(x)}{(x-\mathrm{a})}\) approaches
MCQ+2 / -02023
9Limits Continuity And Differentiability
The function $\mathrm{f}$ defined on \(\left(-\frac{1}{3}, \frac{1}{3}\right)\) by
$$\mathrm{f}(x)=\left\{\begin{array}{cc}
\frac{1}{x} \log \left(\frac{1+3 x}{1-2 x}\right) & , \quad x \neq 0 \\
\mathrm{k} & , \quad x=0
\end{array}\right.$...
$$\mathrm{f}(x)=\left\{\begin{array}{cc}
\frac{1}{x} \log \left(\frac{1+3 x}{1-2 x}\right) & , \quad x \neq 0 \\
\mathrm{k} & , \quad x=0
\end{array}\right.$...
MCQ+2 / -02023
10Limits Continuity And Differentiability
If \(\mathrm{f}(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7\), then \(\lim _\limits{\alpha \rightarrow 0} \frac{f(1-\alpha)-f(1)}{\alpha^3+3 \alpha}=\)
MCQ+2 / -02023
11Limits Continuity And Differentiability
The function \(\mathrm{f}(\mathrm{t})=\frac{1}{\mathrm{t}^2+\mathrm{t}-2}\) where \(\mathrm{t}=\frac{1}{x-1}\) is discontinuous at
MCQ+2 / -02023
12Limits Continuity And Differentiability
If \(f(x)\) is continuous on its domain \([-2,2]\), where
$$f(x)=\left\{\begin{array}{cc}
\frac{\sin a x}{x}+3 & , \text { for }-2 \leq x<0 \\
2 x+7 & , \text { for } 0 \leq x \leq 1 \\
\sqrt{x^2+8}-b & , \text { for } 1< x \leq 2
\end{arra...
$$f(x)=\left\{\begin{array}{cc}
\frac{\sin a x}{x}+3 & , \text { for }-2 \leq x<0 \\
2 x+7 & , \text { for } 0 \leq x \leq 1 \\
\sqrt{x^2+8}-b & , \text { for } 1< x \leq 2
\end{arra...
MCQ+2 / -02023
13Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=\)
MCQ+2 / -02023
14Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 0} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2}\) is
MCQ+2 / -02023
15Limits Continuity And Differentiability
The values of \(a\) and \(b\), so that the function
$$f(x)=\left\{\begin{array}{l}
x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\
2 x \cot x+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\
a \cos 2 x-b \sin x, \frac{\pi}{2} < x \leq \pi
...
$$f(x)=\left\{\begin{array}{l}
x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\
2 x \cot x+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\
a \cos 2 x-b \sin x, \frac{\pi}{2} < x \leq \pi
...
MCQ+2 / -02023
16Limits Continuity And Differentiability
Given $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x<0 \\ \mathrm{a} & , \text { if } x=0 \\ \frac{\sqrt{x}}{\sqrt{16-\sqrt{x}-4}}, & \text { if } x>0\end{array}\right.$$
If \(\mathrm{f}(x)\) is continuous...
If \(\mathrm{f}(x)\) is continuous...
MCQ+2 / -02023
17Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 0} \frac{x \cot 4 x}{\sin ^2 x \cdot \cot ^2(2 x)} \text { is equal to }\)
MCQ+2 / -02023
18Limits Continuity And Differentiability
If $$\mathrm{f}(x)=\left\{\begin{array}{ll}\frac{\sqrt{1+\mathrm{m} x}-\sqrt{1-\mathrm{m} x}}{x}, & -1 \leq x < 0 \\ \frac{2 x+1}{x-2} & , 0 \leq x \leq 1\end{array}\right.$$ is continuous in the interval \([-1,1]\), then \(\mathrm{m}\) is ...
MCQ+2 / -02023
19Limits Continuity And Differentiability
The left-hand derivative of \(\mathrm{f}(x)=[x] \sin (\pi x)\), at \(x=\mathrm{k}, \mathrm{k}\) is an integer and [.] is the greatest integer function, is
MCQ+2 / -02023
20Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2}{x^3-3 x^2+2 x}\right]\) is equal to
MCQ+2 / -02023
21Limits Continuity And Differentiability
If $$\mathrm{f}(x)=\left\{\begin{array}{cc}\frac{x-3}{|x-3|}+\mathrm{a} & , \quad x < 3 \\ \mathrm{a}+\mathrm{b} & , \quad x=3 \\ \frac{|x-3|}{x-3}+\mathrm{b}, & x>3\end{array}\right.$$
Is continuous at \(x=3\), then the value of $$\mathrm{...
Is continuous at \(x=3\), then the value of $$\mathrm{...
MCQ+2 / -02023
22Limits Continuity And Differentiability
\(\text { If } l=\lim _\limits{x \rightarrow 0} \frac{x}{|x|+x^2} \text {, then the value of } l \text { is }\)
MCQ+2 / -02023
23Limits Continuity And Differentiability
If the function \(\mathrm{f}(x)\) is continuous in \(0 \leq x \leq \pi\), then the value of \(2 a+3 b\) is where
$$f(x)= \begin{cases}x+a \sqrt{2} \sin x & \text { if } 0 \leq x < \frac{\pi}{4} \\ 2 x \cot x+b & \text { if } \frac{\pi}{4} \...
$$f(x)= \begin{cases}x+a \sqrt{2} \sin x & \text { if } 0 \leq x < \frac{\pi}{4} \\ 2 x \cot x+b & \text { if } \frac{\pi}{4} \...
MCQ+2 / -02023
24Limits Continuity And Differentiability
The value of \(\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}\) is
MCQ+2 / -02023
25Limits Continuity And Differentiability
\(\lim _\limits{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}=\)
MCQ+2 / -02023
26Limits Continuity And Differentiability
If \(\mathrm{f}(x)=\frac{4}{x^4}\left[1-\cos \frac{x}{2}-\cos \frac{x}{4}+\cos \frac{x}{2} \cdot \cos \frac{x}{4}\right]\) is continuous at \(x=0\), then \(\mathrm{f}(0)\) is
MCQ+2 / -02023
27Limits Continuity And Differentiability
Let \(\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R} / \mathrm{f}(x)=|x-\pi|\left(\mathrm{e}^{|x|}-1\right) \sin |x|\right.\) is not differentiable at \(\mathrm{t}\}\), then \(\mathrm{S}\) is
MCQ+2 / -02023
28Limits Continuity And Differentiability
If \(\lim _\limits{x \rightarrow 1} \frac{x^2-a x+b}{(x-1)}=5\), then \((a+b)\) is equal to
MCQ+2 / -02022
29Limits Continuity And Differentiability
\(\matrix{
{f(x) = a{x^2} + bx + 1,} & {if} & {\left| {2x - 3} \right| \ge 2} \cr
{ = 3x + 2,} & {if} & {{1 \over 2} < x < {5 \over 2}} \cr
}\)
is continuous on its domain, then \(a+b\) has the value
is continuous on its domain, then \(a+b\) has the value
MCQ+2 / -02022